Saturday, December 04, 2021

[eamnctnr] nearby lattice points

we sort 2D, 3D, 4D integer lattice points near the origin by distance from the origin, grouping together points at the same distance.  in the lists at the end of this post, for each distance, we give only a reduced list of representative points whose coordinates are non-negative and non-descending, for brevity.  (note: symmetrically expanding from a reduced list to a full list is nontrivial.)

we also provide some statistics about points all at the same distance:

all-even gives the number of points whose coordinates are all even.  analogously all-odd.  we could also count points with N odd coordinates and dimension-minus-N even coordinates, etc., but this revealed nothing interesting.  4D seems to be the first dimension in which, at a given distance, the parities of the coordinates do not always have the same signature.  for example, this identity underlies the D4 lattice:

4 = 0^2 + 0^2 + 0^2 + 2^2 (all even) = 1^2 + 1^2 + 1^2 + 1^2 (all odd)

if the points split among parity classes in 4D, they split only between the all-even and all-odd classes, and never among the other possible classes (1-odd 3-even, etc.).  the points seem to split if and only if (distance^2) mod 8 = 4 .

in 5D (not explored), we can have other splits:

4 = 0^2 + 0^2 + 0^2 + 0^2 + 2^2 (all even) = 0^2 + 1^2 + 1^2 + 1^2 + 1^2 (1 even, 4 odd)

distance^2 = 25 is the first instance of more than 1 point in the reduced list in 2D.  it is the smallest Pythagorean triple.

25
= 0^2 + 5^2
= 3^2 + 4^2

325 is the first instance of more than 2 points in the reduced list in 2D:

325
= 1^2 + 18^2
= 6^2 + 17^2
= 10^2 + 15^2

3D:
9 = (0,0,3) = (1,2,2)
41 = (0,4,5) = (1,2,6) = (3,4,4).

at greater distances or dimensions, there are many points at a given distance.  here are distances which have a record number of points, subsets of the full lists below.  (the 3D and 4D records cover a greater range, to distance^2 = 399, than the full lists below, which go only to 99.)

2D records

distance-squared 0, record number of points 1, reduced list 1: (0,0)

distance-squared 1, record number of points 4, reduced list 1: (0,1)

distance-squared 5, record number of points 8, reduced list 1: (1,2)

distance-squared 25, record number of points 12, reduced list 2: (0,5); (3,4)

distance-squared 65, record number of points 16, reduced list 2: (1,8); (4,7)

distance-squared 325, record number of points 24, reduced list 3: (1,18); (6,17); (10,15)

3D records

distance-squared 0, record number of points 1, reduced list 1: (0,0,0)

distance-squared 1, record number of points 6, reduced list 1: (0,0,1)

distance-squared 2, record number of points 12, reduced list 1: (0,1,1)

distance-squared 5, record number of points 24, reduced list 1: (0,1,2)

distance-squared 9, record number of points 30, reduced list 2: (0,0,3); (1,2,2)

distance-squared 14, record number of points 48, reduced list 1: (1,2,3)

distance-squared 26, record number of points 72, reduced list 2: (0,1,5); (1,3,4)

distance-squared 41, record number of points 96, reduced list 3: (0,4,5); (1,2,6); (3,4,4)

distance-squared 74, record number of points 120, reduced list 3: (0,5,7); (1,3,8); (3,4,7)

distance-squared 89, record number of points 144, reduced list 4: (0,5,8); (2,2,9); (2,6,7); (3,4,8)

distance-squared 101, record number of points 168, reduced list 4: (0,1,10); (1,6,8); (2,4,9); (4,6,7)

distance-squared 146, record number of points 192, reduced list 5: (0,5,11); (1,1,12); (1,8,9); (3,4,11); (4,7,9)

distance-squared 194, record number of points 240, reduced list 6: (0,5,13); (1,7,12); (3,4,13); (3,8,11); (5,5,12); (7,8,9)

distance-squared 269, record number of points 264, reduced list 6: (0,10,13); (2,3,16); (2,11,12); (3,8,14); (5,10,12); (6,8,13)

distance-squared 314, record number of points 312, reduced list 7: (0,5,17); (1,12,13); (3,4,17); (3,7,16); (5,8,15); (7,11,12); (8,9,13)

distance-squared 341, record number of points 336, reduced list 7: (1,4,18); (1,12,14); (2,9,16); (4,6,17); (4,10,15); (6,7,16); (8,9,14)

4D records

distance-squared 0, record number of points 1, reduced list 1: (0,0,0,0)

distance-squared 1, record number of points 8, reduced list 1: (0,0,0,1)

distance-squared 2, record number of points 24, reduced list 1: (0,0,1,1)

distance-squared 3, record number of points 32, reduced list 1: (0,1,1,1)

distance-squared 5, record number of points 48, reduced list 1: (0,0,1,2)

distance-squared 6, record number of points 96, reduced list 1: (0,1,1,2)

distance-squared 9, record number of points 104, reduced list 2: (0,0,0,3); (0,1,2,2)

distance-squared 10, record number of points 144, reduced list 2: (0,0,1,3); (1,1,2,2)

distance-squared 14, record number of points 192, reduced list 1: (0,1,2,3)

distance-squared 18, record number of points 312, reduced list 3: (0,0,3,3); (0,1,1,4); (1,2,2,3)

distance-squared 26, record number of points 336, reduced list 3: (0,0,1,5); (0,1,3,4); (2,2,3,3)

distance-squared 30, record number of points 576, reduced list 2: (0,1,2,5); (1,2,3,4)

distance-squared 42, record number of points 768, reduced list 4: (0,1,4,5); (1,1,2,6); (1,3,4,4); (2,2,3,5)

distance-squared 54, record number of points 960, reduced list 5: (0,1,2,7); (0,2,5,5); (0,3,3,6); (1,1,4,6); (2,3,4,5)

distance-squared 66, record number of points 1152, reduced list 6: (0,1,1,8); (0,1,4,7); (0,4,5,5); (1,2,5,6); (2,2,3,7); (3,4,4,5)

distance-squared 78, record number of points 1344, reduced list 4: (0,2,5,7); (1,2,3,8); (1,4,5,6); (2,3,4,7)

distance-squared 90, record number of points 1872, reduced list 9: (0,0,3,9); (0,1,5,8); (0,4,5,7); (1,2,2,9); (1,2,6,7); (1,3,4,8); (2,5,5,6); (3,3,6,6); (3,4,4,7)

distance-squared 114, record number of points 1920, reduced list 8: (0,1,7,8); (0,4,7,7); (0,5,5,8); (1,2,3,10); (1,4,4,9); (2,2,5,9); (2,5,6,7); (3,4,5,8)

distance-squared 126, record number of points 2496, reduced list 8: (0,1,2,11); (0,1,5,10); (0,3,6,9); (1,3,4,10); (1,5,6,8); (2,3,7,8); (2,4,5,9); (4,5,6,7)

distance-squared 150, record number of points 2976, reduced list 11: (0,1,7,10); (0,2,5,11); (0,5,5,10); (1,1,2,12); (1,2,8,9); (1,6,7,8); (2,3,4,11); (2,4,7,9); (3,4,5,10); (4,6,7,7); (5,5,6,8)

distance-squared 186, record number of points 3072, reduced list 11: (0,1,4,13); (0,1,8,11); (0,4,7,11); (1,2,9,10); (1,4,5,12); (1,6,7,10); (2,2,3,13); (2,5,6,11); (3,7,8,8); (4,5,8,9); (5,5,6,10)

distance-squared 198, record number of points 3744, reduced list 14: (0,1,1,14); (0,2,5,13); (0,6,9,9); (0,7,7,10); (1,2,7,12); (1,4,9,10); (2,3,4,13); (2,3,8,11); (2,5,5,12); (2,7,8,9); (3,3,6,12); (3,5,8,10); (4,5,6,11); (6,7,7,8)

distance-squared 210, record number of points 4608, reduced list 16: (0,4,5,13); (0,5,8,11); (1,1,8,12); (1,2,3,14); (1,2,6,13); (1,3,10,10); (1,4,7,12); (1,8,8,9); (2,2,9,11); (2,5,9,10); (2,6,7,11); (3,4,4,13); (3,4,8,11); (4,5,5,12); (4,7,8,9); (5,6,7,10)

distance-squared 270, record number of points 5760, reduced list 17: (0,1,10,13); (0,3,6,15); (0,5,7,14); (0,7,10,11); (1,2,3,16); (1,2,11,12); (1,3,8,14); (1,5,10,12); (1,6,8,13); (2,4,5,15); (2,4,9,13); (2,8,9,11); (3,4,7,14); (3,6,9,12); (4,6,7,13); (5,8,9,10); (6,7,8,11)

distance-squared 330, record number of points 6912, reduced list 22: (0,4,5,17); (0,5,7,16); (1,1,2,18); (1,2,6,17); (1,2,10,15); (1,3,8,16); (1,4,12,13); (1,8,11,12); (2,3,11,14); (2,6,11,13); (2,7,9,14); (3,4,4,17); (3,4,7,16); (3,5,10,14); (3,10,10,11); (4,5,8,15); (4,7,11,12); (4,8,9,13); (5,6,10,13); (6,7,7,14); (7,9,10,10); (8,8,9,11)

distance-squared 378, record number of points 7680, reduced list 28: (0,1,4,19); (0,1,11,16); (0,3,12,15); (0,5,8,17); (1,2,7,18); (1,8,12,13); (1,9,10,14); (2,2,3,19); (2,2,9,17); (2,3,13,14); (2,5,5,18); (2,6,7,17); (2,6,13,13); (2,7,10,15); (3,3,6,18); (3,4,8,17); (3,7,8,16); (3,9,12,12); (3,10,10,13); (4,4,11,15); (4,5,9,16); (4,7,12,13); (5,6,11,14); (5,8,8,15); (6,6,9,15); (6,10,11,11); (7,8,11,12); (8,8,9,13)

distance-squared 390, record number of points 8064, reduced list 26: (0,1,10,17); (0,2,5,19); (0,5,13,14); (0,10,11,13); (1,1,8,18); (1,4,7,18); (1,6,8,17); (1,7,12,14); (1,8,10,15); (2,3,4,19); (2,3,11,16); (2,4,9,17); (2,7,9,16); (2,11,11,12); (3,4,13,14); (3,5,10,16); (3,8,11,14); (4,5,5,18); (4,6,7,17); (4,6,13,13); (4,7,10,15); (5,5,12,14); (5,10,11,12); (6,7,7,16); (6,8,11,13); (7,8,9,14)

in 4D, every non-negative integer is possible as a squared distance.

inspiration: random graphs in which each point is connected to a random subset of points not too far away from it.

digression: consider drawing (embedding) on a 2D plane such a graph from 3D or higher dimension.  note that the graph is infinite.  assume, for readability, nodes as drawn on the plane must all be separated by a minimum distance, but there is otherwise free placement.  can the infinite graph be drawn bounding the length of edges?  I suspect it cannot.


full lists:

2D

distance-squared 0, number of points 1, running total 1, all-even 1, all-odd 0, reduced list 1: (0,0)

distance-squared 1, number of points 4, running total 5, all-even 0, all-odd 0, reduced list 1: (0,1)

distance-squared 2, number of points 4, running total 9, all-even 0, all-odd 4, reduced list 1: (1,1)

distance-squared 4, number of points 4, running total 13, all-even 4, all-odd 0, reduced list 1: (0,2)

distance-squared 5, number of points 8, running total 21, all-even 0, all-odd 0, reduced list 1: (1,2)

distance-squared 8, number of points 4, running total 25, all-even 4, all-odd 0, reduced list 1: (2,2)

distance-squared 9, number of points 4, running total 29, all-even 0, all-odd 0, reduced list 1: (0,3)

distance-squared 10, number of points 8, running total 37, all-even 0, all-odd 8, reduced list 1: (1,3)

distance-squared 13, number of points 8, running total 45, all-even 0, all-odd 0, reduced list 1: (2,3)

distance-squared 16, number of points 4, running total 49, all-even 4, all-odd 0, reduced list 1: (0,4)

distance-squared 17, number of points 8, running total 57, all-even 0, all-odd 0, reduced list 1: (1,4)

distance-squared 18, number of points 4, running total 61, all-even 0, all-odd 4, reduced list 1: (3,3)

distance-squared 20, number of points 8, running total 69, all-even 8, all-odd 0, reduced list 1: (2,4)

distance-squared 25, number of points 12, running total 81, all-even 0, all-odd 0, reduced list 2: (0,5); (3,4)

distance-squared 26, number of points 8, running total 89, all-even 0, all-odd 8, reduced list 1: (1,5)

distance-squared 29, number of points 8, running total 97, all-even 0, all-odd 0, reduced list 1: (2,5)

distance-squared 32, number of points 4, running total 101, all-even 4, all-odd 0, reduced list 1: (4,4)

distance-squared 34, number of points 8, running total 109, all-even 0, all-odd 8, reduced list 1: (3,5)

distance-squared 36, number of points 4, running total 113, all-even 4, all-odd 0, reduced list 1: (0,6)

distance-squared 37, number of points 8, running total 121, all-even 0, all-odd 0, reduced list 1: (1,6)

distance-squared 40, number of points 8, running total 129, all-even 8, all-odd 0, reduced list 1: (2,6)

distance-squared 41, number of points 8, running total 137, all-even 0, all-odd 0, reduced list 1: (4,5)

distance-squared 45, number of points 8, running total 145, all-even 0, all-odd 0, reduced list 1: (3,6)

distance-squared 49, number of points 4, running total 149, all-even 0, all-odd 0, reduced list 1: (0,7)

distance-squared 50, number of points 12, running total 161, all-even 0, all-odd 12, reduced list 2: (1,7); (5,5)

distance-squared 52, number of points 8, running total 169, all-even 8, all-odd 0, reduced list 1: (4,6)

distance-squared 53, number of points 8, running total 177, all-even 0, all-odd 0, reduced list 1: (2,7)

distance-squared 58, number of points 8, running total 185, all-even 0, all-odd 8, reduced list 1: (3,7)

distance-squared 61, number of points 8, running total 193, all-even 0, all-odd 0, reduced list 1: (5,6)

distance-squared 64, number of points 4, running total 197, all-even 4, all-odd 0, reduced list 1: (0,8)

distance-squared 65, number of points 16, running total 213, all-even 0, all-odd 0, reduced list 2: (1,8); (4,7)

distance-squared 68, number of points 8, running total 221, all-even 8, all-odd 0, reduced list 1: (2,8)

distance-squared 72, number of points 4, running total 225, all-even 4, all-odd 0, reduced list 1: (6,6)

distance-squared 73, number of points 8, running total 233, all-even 0, all-odd 0, reduced list 1: (3,8)

distance-squared 74, number of points 8, running total 241, all-even 0, all-odd 8, reduced list 1: (5,7)

distance-squared 80, number of points 8, running total 249, all-even 8, all-odd 0, reduced list 1: (4,8)

distance-squared 81, number of points 4, running total 253, all-even 0, all-odd 0, reduced list 1: (0,9)

distance-squared 82, number of points 8, running total 261, all-even 0, all-odd 8, reduced list 1: (1,9)

distance-squared 85, number of points 16, running total 277, all-even 0, all-odd 0, reduced list 2: (2,9); (6,7)

distance-squared 89, number of points 8, running total 285, all-even 0, all-odd 0, reduced list 1: (5,8)

distance-squared 90, number of points 8, running total 293, all-even 0, all-odd 8, reduced list 1: (3,9)

distance-squared 97, number of points 8, running total 301, all-even 0, all-odd 0, reduced list 1: (4,9)

distance-squared 98, number of points 4, running total 305, all-even 0, all-odd 4, reduced list 1: (7,7)

distance-squared 100, number of points 12, running total 317, all-even 12, all-odd 0, reduced list 2: (0,10); (6,8)

distance-squared 101, number of points 8, running total 325, all-even 0, all-odd 0, reduced list 1: (1,10)

distance-squared 104, number of points 8, running total 333, all-even 8, all-odd 0, reduced list 1: (2,10)

distance-squared 106, number of points 8, running total 341, all-even 0, all-odd 8, reduced list 1: (5,9)

distance-squared 109, number of points 8, running total 349, all-even 0, all-odd 0, reduced list 1: (3,10)

distance-squared 113, number of points 8, running total 357, all-even 0, all-odd 0, reduced list 1: (7,8)

distance-squared 116, number of points 8, running total 365, all-even 8, all-odd 0, reduced list 1: (4,10)

distance-squared 117, number of points 8, running total 373, all-even 0, all-odd 0, reduced list 1: (6,9)

distance-squared 121, number of points 4, running total 377, all-even 0, all-odd 0, reduced list 1: (0,11)

distance-squared 122, number of points 8, running total 385, all-even 0, all-odd 8, reduced list 1: (1,11)

distance-squared 125, number of points 16, running total 401, all-even 0, all-odd 0, reduced list 2: (2,11); (5,10)

distance-squared 128, number of points 4, running total 405, all-even 4, all-odd 0, reduced list 1: (8,8)

distance-squared 130, number of points 16, running total 421, all-even 0, all-odd 16, reduced list 2: (3,11); (7,9)

distance-squared 136, number of points 8, running total 429, all-even 8, all-odd 0, reduced list 1: (6,10)

distance-squared 137, number of points 8, running total 437, all-even 0, all-odd 0, reduced list 1: (4,11)

distance-squared 144, number of points 4, running total 441, all-even 4, all-odd 0, reduced list 1: (0,12)

distance-squared 145, number of points 16, running total 457, all-even 0, all-odd 0, reduced list 2: (1,12); (8,9)

distance-squared 146, number of points 8, running total 465, all-even 0, all-odd 8, reduced list 1: (5,11)

distance-squared 148, number of points 8, running total 473, all-even 8, all-odd 0, reduced list 1: (2,12)

distance-squared 149, number of points 8, running total 481, all-even 0, all-odd 0, reduced list 1: (7,10)

distance-squared 153, number of points 8, running total 489, all-even 0, all-odd 0, reduced list 1: (3,12)

distance-squared 157, number of points 8, running total 497, all-even 0, all-odd 0, reduced list 1: (6,11)

distance-squared 160, number of points 8, running total 505, all-even 8, all-odd 0, reduced list 1: (4,12)

distance-squared 162, number of points 4, running total 509, all-even 0, all-odd 4, reduced list 1: (9,9)

distance-squared 164, number of points 8, running total 517, all-even 8, all-odd 0, reduced list 1: (8,10)

distance-squared 169, number of points 12, running total 529, all-even 0, all-odd 0, reduced list 2: (0,13); (5,12)

distance-squared 170, number of points 16, running total 545, all-even 0, all-odd 16, reduced list 2: (1,13); (7,11)

distance-squared 173, number of points 8, running total 553, all-even 0, all-odd 0, reduced list 1: (2,13)

distance-squared 178, number of points 8, running total 561, all-even 0, all-odd 8, reduced list 1: (3,13)

distance-squared 180, number of points 8, running total 569, all-even 8, all-odd 0, reduced list 1: (6,12)

distance-squared 181, number of points 8, running total 577, all-even 0, all-odd 0, reduced list 1: (9,10)

distance-squared 185, number of points 16, running total 593, all-even 0, all-odd 0, reduced list 2: (4,13); (8,11)

distance-squared 193, number of points 8, running total 601, all-even 0, all-odd 0, reduced list 1: (7,12)

distance-squared 194, number of points 8, running total 609, all-even 0, all-odd 8, reduced list 1: (5,13)

distance-squared 196, number of points 4, running total 613, all-even 4, all-odd 0, reduced list 1: (0,14)

distance-squared 197, number of points 8, running total 621, all-even 0, all-odd 0, reduced list 1: (1,14)

distance-squared 200, number of points 12, running total 633, all-even 12, all-odd 0, reduced list 2: (2,14); (10,10)

distance-squared 202, number of points 8, running total 641, all-even 0, all-odd 8, reduced list 1: (9,11)

distance-squared 205, number of points 16, running total 657, all-even 0, all-odd 0, reduced list 2: (3,14); (6,13)

distance-squared 208, number of points 8, running total 665, all-even 8, all-odd 0, reduced list 1: (8,12)

distance-squared 212, number of points 8, running total 673, all-even 8, all-odd 0, reduced list 1: (4,14)

distance-squared 218, number of points 8, running total 681, all-even 0, all-odd 8, reduced list 1: (7,13)

distance-squared 221, number of points 16, running total 697, all-even 0, all-odd 0, reduced list 2: (5,14); (10,11)

distance-squared 225, number of points 12, running total 709, all-even 0, all-odd 0, reduced list 2: (0,15); (9,12)

distance-squared 226, number of points 8, running total 717, all-even 0, all-odd 8, reduced list 1: (1,15)

distance-squared 229, number of points 8, running total 725, all-even 0, all-odd 0, reduced list 1: (2,15)

distance-squared 232, number of points 8, running total 733, all-even 8, all-odd 0, reduced list 1: (6,14)

distance-squared 233, number of points 8, running total 741, all-even 0, all-odd 0, reduced list 1: (8,13)

distance-squared 234, number of points 8, running total 749, all-even 0, all-odd 8, reduced list 1: (3,15)

distance-squared 241, number of points 8, running total 757, all-even 0, all-odd 0, reduced list 1: (4,15)

distance-squared 242, number of points 4, running total 761, all-even 0, all-odd 4, reduced list 1: (11,11)

distance-squared 244, number of points 8, running total 769, all-even 8, all-odd 0, reduced list 1: (10,12)

distance-squared 245, number of points 8, running total 777, all-even 0, all-odd 0, reduced list 1: (7,14)

distance-squared 250, number of points 16, running total 793, all-even 0, all-odd 16, reduced list 2: (5,15); (9,13)

distance-squared 256, number of points 4, running total 797, all-even 4, all-odd 0, reduced list 1: (0,16)

distance-squared 257, number of points 8, running total 805, all-even 0, all-odd 0, reduced list 1: (1,16)

distance-squared 260, number of points 16, running total 821, all-even 16, all-odd 0, reduced list 2: (2,16); (8,14)

distance-squared 261, number of points 8, running total 829, all-even 0, all-odd 0, reduced list 1: (6,15)

distance-squared 265, number of points 16, running total 845, all-even 0, all-odd 0, reduced list 2: (3,16); (11,12)

distance-squared 269, number of points 8, running total 853, all-even 0, all-odd 0, reduced list 1: (10,13)

distance-squared 272, number of points 8, running total 861, all-even 8, all-odd 0, reduced list 1: (4,16)

distance-squared 274, number of points 8, running total 869, all-even 0, all-odd 8, reduced list 1: (7,15)

distance-squared 277, number of points 8, running total 877, all-even 0, all-odd 0, reduced list 1: (9,14)

distance-squared 281, number of points 8, running total 885, all-even 0, all-odd 0, reduced list 1: (5,16)

distance-squared 288, number of points 4, running total 889, all-even 4, all-odd 0, reduced list 1: (12,12)

distance-squared 289, number of points 12, running total 901, all-even 0, all-odd 0, reduced list 2: (0,17); (8,15)

distance-squared 290, number of points 16, running total 917, all-even 0, all-odd 16, reduced list 2: (1,17); (11,13)

distance-squared 292, number of points 8, running total 925, all-even 8, all-odd 0, reduced list 1: (6,16)

distance-squared 293, number of points 8, running total 933, all-even 0, all-odd 0, reduced list 1: (2,17)

distance-squared 296, number of points 8, running total 941, all-even 8, all-odd 0, reduced list 1: (10,14)

distance-squared 298, number of points 8, running total 949, all-even 0, all-odd 8, reduced list 1: (3,17)

distance-squared 305, number of points 16, running total 965, all-even 0, all-odd 0, reduced list 2: (4,17); (7,16)

distance-squared 306, number of points 8, running total 973, all-even 0, all-odd 8, reduced list 1: (9,15)

distance-squared 313, number of points 8, running total 981, all-even 0, all-odd 0, reduced list 1: (12,13)

distance-squared 314, number of points 8, running total 989, all-even 0, all-odd 8, reduced list 1: (5,17)

distance-squared 317, number of points 8, running total 997, all-even 0, all-odd 0, reduced list 1: (11,14)

distance-squared 320, number of points 8, running total 1005, all-even 8, all-odd 0, reduced list 1: (8,16)

distance-squared 324, number of points 4, running total 1009, all-even 4, all-odd 0, reduced list 1: (0,18)

distance-squared 325, number of points 24, running total 1033, all-even 0, all-odd 0, reduced list 3: (1,18); (6,17); (10,15)

distance-squared 328, number of points 8, running total 1041, all-even 8, all-odd 0, reduced list 1: (2,18)

distance-squared 333, number of points 8, running total 1049, all-even 0, all-odd 0, reduced list 1: (3,18)

distance-squared 337, number of points 8, running total 1057, all-even 0, all-odd 0, reduced list 1: (9,16)

distance-squared 338, number of points 12, running total 1069, all-even 0, all-odd 12, reduced list 2: (7,17); (13,13)

distance-squared 340, number of points 16, running total 1085, all-even 16, all-odd 0, reduced list 2: (4,18); (12,14)

distance-squared 346, number of points 8, running total 1093, all-even 0, all-odd 8, reduced list 1: (11,15)

distance-squared 349, number of points 8, running total 1101, all-even 0, all-odd 0, reduced list 1: (5,18)

distance-squared 353, number of points 8, running total 1109, all-even 0, all-odd 0, reduced list 1: (8,17)

distance-squared 356, number of points 8, running total 1117, all-even 8, all-odd 0, reduced list 1: (10,16)

distance-squared 360, number of points 8, running total 1125, all-even 8, all-odd 0, reduced list 1: (6,18)

distance-squared 361, number of points 4, running total 1129, all-even 0, all-odd 0, reduced list 1: (0,19)

distance-squared 362, number of points 8, running total 1137, all-even 0, all-odd 8, reduced list 1: (1,19)

distance-squared 365, number of points 16, running total 1153, all-even 0, all-odd 0, reduced list 2: (2,19); (13,14)

distance-squared 369, number of points 8, running total 1161, all-even 0, all-odd 0, reduced list 1: (12,15)

distance-squared 370, number of points 16, running total 1177, all-even 0, all-odd 16, reduced list 2: (3,19); (9,17)

distance-squared 373, number of points 8, running total 1185, all-even 0, all-odd 0, reduced list 1: (7,18)

distance-squared 377, number of points 16, running total 1201, all-even 0, all-odd 0, reduced list 2: (4,19); (11,16)

distance-squared 386, number of points 8, running total 1209, all-even 0, all-odd 8, reduced list 1: (5,19)

distance-squared 388, number of points 8, running total 1217, all-even 8, all-odd 0, reduced list 1: (8,18)

distance-squared 389, number of points 8, running total 1225, all-even 0, all-odd 0, reduced list 1: (10,17)

distance-squared 392, number of points 4, running total 1229, all-even 4, all-odd 0, reduced list 1: (14,14)

distance-squared 394, number of points 8, running total 1237, all-even 0, all-odd 8, reduced list 1: (13,15)

distance-squared 397, number of points 8, running total 1245, all-even 0, all-odd 0, reduced list 1: (6,19)

3D

distance-squared 0, number of points 1, running total 1, all-even 1, all-odd 0, reduced list 1: (0,0,0)

distance-squared 1, number of points 6, running total 7, all-even 0, all-odd 0, reduced list 1: (0,0,1)

distance-squared 2, number of points 12, running total 19, all-even 0, all-odd 0, reduced list 1: (0,1,1)

distance-squared 3, number of points 8, running total 27, all-even 0, all-odd 8, reduced list 1: (1,1,1)

distance-squared 4, number of points 6, running total 33, all-even 6, all-odd 0, reduced list 1: (0,0,2)

distance-squared 5, number of points 24, running total 57, all-even 0, all-odd 0, reduced list 1: (0,1,2)

distance-squared 6, number of points 24, running total 81, all-even 0, all-odd 0, reduced list 1: (1,1,2)

distance-squared 8, number of points 12, running total 93, all-even 12, all-odd 0, reduced list 1: (0,2,2)

distance-squared 9, number of points 30, running total 123, all-even 0, all-odd 0, reduced list 2: (0,0,3); (1,2,2)

distance-squared 10, number of points 24, running total 147, all-even 0, all-odd 0, reduced list 1: (0,1,3)

distance-squared 11, number of points 24, running total 171, all-even 0, all-odd 24, reduced list 1: (1,1,3)

distance-squared 12, number of points 8, running total 179, all-even 8, all-odd 0, reduced list 1: (2,2,2)

distance-squared 13, number of points 24, running total 203, all-even 0, all-odd 0, reduced list 1: (0,2,3)

distance-squared 14, number of points 48, running total 251, all-even 0, all-odd 0, reduced list 1: (1,2,3)

distance-squared 16, number of points 6, running total 257, all-even 6, all-odd 0, reduced list 1: (0,0,4)

distance-squared 17, number of points 48, running total 305, all-even 0, all-odd 0, reduced list 2: (0,1,4); (2,2,3)

distance-squared 18, number of points 36, running total 341, all-even 0, all-odd 0, reduced list 2: (0,3,3); (1,1,4)

distance-squared 19, number of points 24, running total 365, all-even 0, all-odd 24, reduced list 1: (1,3,3)

distance-squared 20, number of points 24, running total 389, all-even 24, all-odd 0, reduced list 1: (0,2,4)

distance-squared 21, number of points 48, running total 437, all-even 0, all-odd 0, reduced list 1: (1,2,4)

distance-squared 22, number of points 24, running total 461, all-even 0, all-odd 0, reduced list 1: (2,3,3)

distance-squared 24, number of points 24, running total 485, all-even 24, all-odd 0, reduced list 1: (2,2,4)

distance-squared 25, number of points 30, running total 515, all-even 0, all-odd 0, reduced list 2: (0,0,5); (0,3,4)

distance-squared 26, number of points 72, running total 587, all-even 0, all-odd 0, reduced list 2: (0,1,5); (1,3,4)

distance-squared 27, number of points 32, running total 619, all-even 0, all-odd 32, reduced list 2: (1,1,5); (3,3,3)

distance-squared 29, number of points 72, running total 691, all-even 0, all-odd 0, reduced list 2: (0,2,5); (2,3,4)

distance-squared 30, number of points 48, running total 739, all-even 0, all-odd 0, reduced list 1: (1,2,5)

distance-squared 32, number of points 12, running total 751, all-even 12, all-odd 0, reduced list 1: (0,4,4)

distance-squared 33, number of points 48, running total 799, all-even 0, all-odd 0, reduced list 2: (1,4,4); (2,2,5)

distance-squared 34, number of points 48, running total 847, all-even 0, all-odd 0, reduced list 2: (0,3,5); (3,3,4)

distance-squared 35, number of points 48, running total 895, all-even 0, all-odd 48, reduced list 1: (1,3,5)

distance-squared 36, number of points 30, running total 925, all-even 30, all-odd 0, reduced list 2: (0,0,6); (2,4,4)

distance-squared 37, number of points 24, running total 949, all-even 0, all-odd 0, reduced list 1: (0,1,6)

distance-squared 38, number of points 72, running total 1021, all-even 0, all-odd 0, reduced list 2: (1,1,6); (2,3,5)

distance-squared 40, number of points 24, running total 1045, all-even 24, all-odd 0, reduced list 1: (0,2,6)

distance-squared 41, number of points 96, running total 1141, all-even 0, all-odd 0, reduced list 3: (0,4,5); (1,2,6); (3,4,4)

distance-squared 42, number of points 48, running total 1189, all-even 0, all-odd 0, reduced list 1: (1,4,5)

distance-squared 43, number of points 24, running total 1213, all-even 0, all-odd 24, reduced list 1: (3,3,5)

distance-squared 44, number of points 24, running total 1237, all-even 24, all-odd 0, reduced list 1: (2,2,6)

distance-squared 45, number of points 72, running total 1309, all-even 0, all-odd 0, reduced list 2: (0,3,6); (2,4,5)

distance-squared 46, number of points 48, running total 1357, all-even 0, all-odd 0, reduced list 1: (1,3,6)

distance-squared 48, number of points 8, running total 1365, all-even 8, all-odd 0, reduced list 1: (4,4,4)

distance-squared 49, number of points 54, running total 1419, all-even 0, all-odd 0, reduced list 2: (0,0,7); (2,3,6)

distance-squared 50, number of points 84, running total 1503, all-even 0, all-odd 0, reduced list 3: (0,1,7); (0,5,5); (3,4,5)

distance-squared 51, number of points 48, running total 1551, all-even 0, all-odd 48, reduced list 2: (1,1,7); (1,5,5)

distance-squared 52, number of points 24, running total 1575, all-even 24, all-odd 0, reduced list 1: (0,4,6)

distance-squared 53, number of points 72, running total 1647, all-even 0, all-odd 0, reduced list 2: (0,2,7); (1,4,6)

distance-squared 54, number of points 96, running total 1743, all-even 0, all-odd 0, reduced list 3: (1,2,7); (2,5,5); (3,3,6)

distance-squared 56, number of points 48, running total 1791, all-even 48, all-odd 0, reduced list 1: (2,4,6)

distance-squared 57, number of points 48, running total 1839, all-even 0, all-odd 0, reduced list 2: (2,2,7); (4,4,5)

distance-squared 58, number of points 24, running total 1863, all-even 0, all-odd 0, reduced list 1: (0,3,7)

distance-squared 59, number of points 72, running total 1935, all-even 0, all-odd 72, reduced list 2: (1,3,7); (3,5,5)

distance-squared 61, number of points 72, running total 2007, all-even 0, all-odd 0, reduced list 2: (0,5,6); (3,4,6)

distance-squared 62, number of points 96, running total 2103, all-even 0, all-odd 0, reduced list 2: (1,5,6); (2,3,7)

distance-squared 64, number of points 6, running total 2109, all-even 6, all-odd 0, reduced list 1: (0,0,8)

distance-squared 65, number of points 96, running total 2205, all-even 0, all-odd 0, reduced list 3: (0,1,8); (0,4,7); (2,5,6)

distance-squared 66, number of points 96, running total 2301, all-even 0, all-odd 0, reduced list 3: (1,1,8); (1,4,7); (4,5,5)

distance-squared 67, number of points 24, running total 2325, all-even 0, all-odd 24, reduced list 1: (3,3,7)

distance-squared 68, number of points 48, running total 2373, all-even 48, all-odd 0, reduced list 2: (0,2,8); (4,4,6)

distance-squared 69, number of points 96, running total 2469, all-even 0, all-odd 0, reduced list 2: (1,2,8); (2,4,7)

distance-squared 70, number of points 48, running total 2517, all-even 0, all-odd 0, reduced list 1: (3,5,6)

distance-squared 72, number of points 36, running total 2553, all-even 36, all-odd 0, reduced list 2: (0,6,6); (2,2,8)

distance-squared 73, number of points 48, running total 2601, all-even 0, all-odd 0, reduced list 2: (0,3,8); (1,6,6)

distance-squared 74, number of points 120, running total 2721, all-even 0, all-odd 0, reduced list 3: (0,5,7); (1,3,8); (3,4,7)

distance-squared 75, number of points 56, running total 2777, all-even 0, all-odd 56, reduced list 2: (1,5,7); (5,5,5)

distance-squared 76, number of points 24, running total 2801, all-even 24, all-odd 0, reduced list 1: (2,6,6)

distance-squared 77, number of points 96, running total 2897, all-even 0, all-odd 0, reduced list 2: (2,3,8); (4,5,6)

distance-squared 78, number of points 48, running total 2945, all-even 0, all-odd 0, reduced list 1: (2,5,7)

distance-squared 80, number of points 24, running total 2969, all-even 24, all-odd 0, reduced list 1: (0,4,8)

distance-squared 81, number of points 102, running total 3071, all-even 0, all-odd 0, reduced list 4: (0,0,9); (1,4,8); (3,6,6); (4,4,7)

distance-squared 82, number of points 48, running total 3119, all-even 0, all-odd 0, reduced list 2: (0,1,9); (3,3,8)

distance-squared 83, number of points 72, running total 3191, all-even 0, all-odd 72, reduced list 2: (1,1,9); (3,5,7)

distance-squared 84, number of points 48, running total 3239, all-even 48, all-odd 0, reduced list 1: (2,4,8)

distance-squared 85, number of points 48, running total 3287, all-even 0, all-odd 0, reduced list 2: (0,2,9); (0,6,7)

distance-squared 86, number of points 120, running total 3407, all-even 0, all-odd 0, reduced list 3: (1,2,9); (1,6,7); (5,5,6)

distance-squared 88, number of points 24, running total 3431, all-even 24, all-odd 0, reduced list 1: (4,6,6)

distance-squared 89, number of points 144, running total 3575, all-even 0, all-odd 0, reduced list 4: (0,5,8); (2,2,9); (2,6,7); (3,4,8)

distance-squared 90, number of points 120, running total 3695, all-even 0, all-odd 0, reduced list 3: (0,3,9); (1,5,8); (4,5,7)

distance-squared 91, number of points 48, running total 3743, all-even 0, all-odd 48, reduced list 1: (1,3,9)

distance-squared 93, number of points 48, running total 3791, all-even 0, all-odd 0, reduced list 1: (2,5,8)

distance-squared 94, number of points 96, running total 3887, all-even 0, all-odd 0, reduced list 2: (2,3,9); (3,6,7)

distance-squared 96, number of points 24, running total 3911, all-even 24, all-odd 0, reduced list 1: (4,4,8)

distance-squared 97, number of points 48, running total 3959, all-even 0, all-odd 0, reduced list 2: (0,4,9); (5,6,6)

distance-squared 98, number of points 108, running total 4067, all-even 0, all-odd 0, reduced list 3: (0,7,7); (1,4,9); (3,5,8)

distance-squared 99, number of points 72, running total 4139, all-even 0, all-odd 72, reduced list 3: (1,7,7); (3,3,9); (5,5,7)

4D

distance-squared 0, number of points 1, running total 1, all-even 1, all-odd 0, reduced list 1: (0,0,0,0)

distance-squared 1, number of points 8, running total 9, all-even 0, all-odd 0, reduced list 1: (0,0,0,1)

distance-squared 2, number of points 24, running total 33, all-even 0, all-odd 0, reduced list 1: (0,0,1,1)

distance-squared 3, number of points 32, running total 65, all-even 0, all-odd 0, reduced list 1: (0,1,1,1)

distance-squared 4, number of points 24, running total 89, all-even 8, all-odd 16, reduced list 2: (0,0,0,2); (1,1,1,1)

distance-squared 5, number of points 48, running total 137, all-even 0, all-odd 0, reduced list 1: (0,0,1,2)

distance-squared 6, number of points 96, running total 233, all-even 0, all-odd 0, reduced list 1: (0,1,1,2)

distance-squared 7, number of points 64, running total 297, all-even 0, all-odd 0, reduced list 1: (1,1,1,2)

distance-squared 8, number of points 24, running total 321, all-even 24, all-odd 0, reduced list 1: (0,0,2,2)

distance-squared 9, number of points 104, running total 425, all-even 0, all-odd 0, reduced list 2: (0,0,0,3); (0,1,2,2)

distance-squared 10, number of points 144, running total 569, all-even 0, all-odd 0, reduced list 2: (0,0,1,3); (1,1,2,2)

distance-squared 11, number of points 96, running total 665, all-even 0, all-odd 0, reduced list 1: (0,1,1,3)

distance-squared 12, number of points 96, running total 761, all-even 32, all-odd 64, reduced list 2: (0,2,2,2); (1,1,1,3)

distance-squared 13, number of points 112, running total 873, all-even 0, all-odd 0, reduced list 2: (0,0,2,3); (1,2,2,2)

distance-squared 14, number of points 192, running total 1065, all-even 0, all-odd 0, reduced list 1: (0,1,2,3)

distance-squared 15, number of points 192, running total 1257, all-even 0, all-odd 0, reduced list 1: (1,1,2,3)

distance-squared 16, number of points 24, running total 1281, all-even 24, all-odd 0, reduced list 2: (0,0,0,4); (2,2,2,2)

distance-squared 17, number of points 144, running total 1425, all-even 0, all-odd 0, reduced list 2: (0,0,1,4); (0,2,2,3)

distance-squared 18, number of points 312, running total 1737, all-even 0, all-odd 0, reduced list 3: (0,0,3,3); (0,1,1,4); (1,2,2,3)

distance-squared 19, number of points 160, running total 1897, all-even 0, all-odd 0, reduced list 2: (0,1,3,3); (1,1,1,4)

distance-squared 20, number of points 144, running total 2041, all-even 48, all-odd 96, reduced list 2: (0,0,2,4); (1,1,3,3)

distance-squared 21, number of points 256, running total 2297, all-even 0, all-odd 0, reduced list 2: (0,1,2,4); (2,2,2,3)

distance-squared 22, number of points 288, running total 2585, all-even 0, all-odd 0, reduced list 2: (0,2,3,3); (1,1,2,4)

distance-squared 23, number of points 192, running total 2777, all-even 0, all-odd 0, reduced list 1: (1,2,3,3)

distance-squared 24, number of points 96, running total 2873, all-even 96, all-odd 0, reduced list 1: (0,2,2,4)

distance-squared 25, number of points 248, running total 3121, all-even 0, all-odd 0, reduced list 3: (0,0,0,5); (0,0,3,4); (1,2,2,4)

distance-squared 26, number of points 336, running total 3457, all-even 0, all-odd 0, reduced list 3: (0,0,1,5); (0,1,3,4); (2,2,3,3)

distance-squared 27, number of points 320, running total 3777, all-even 0, all-odd 0, reduced list 3: (0,1,1,5); (0,3,3,3); (1,1,3,4)

distance-squared 28, number of points 192, running total 3969, all-even 64, all-odd 128, reduced list 3: (1,1,1,5); (1,3,3,3); (2,2,2,4)

distance-squared 29, number of points 240, running total 4209, all-even 0, all-odd 0, reduced list 2: (0,0,2,5); (0,2,3,4)

distance-squared 30, number of points 576, running total 4785, all-even 0, all-odd 0, reduced list 2: (0,1,2,5); (1,2,3,4)

distance-squared 31, number of points 256, running total 5041, all-even 0, all-odd 0, reduced list 2: (1,1,2,5); (2,3,3,3)

distance-squared 32, number of points 24, running total 5065, all-even 24, all-odd 0, reduced list 1: (0,0,4,4)

distance-squared 33, number of points 384, running total 5449, all-even 0, all-odd 0, reduced list 3: (0,1,4,4); (0,2,2,5); (2,2,3,4)

distance-squared 34, number of points 432, running total 5881, all-even 0, all-odd 0, reduced list 4: (0,0,3,5); (0,3,3,4); (1,1,4,4); (1,2,2,5)

distance-squared 35, number of points 384, running total 6265, all-even 0, all-odd 0, reduced list 2: (0,1,3,5); (1,3,3,4)

distance-squared 36, number of points 312, running total 6577, all-even 104, all-odd 208, reduced list 4: (0,0,0,6); (0,2,4,4); (1,1,3,5); (3,3,3,3)

distance-squared 37, number of points 304, running total 6881, all-even 0, all-odd 0, reduced list 3: (0,0,1,6); (1,2,4,4); (2,2,2,5)

distance-squared 38, number of points 480, running total 7361, all-even 0, all-odd 0, reduced list 3: (0,1,1,6); (0,2,3,5); (2,3,3,4)

distance-squared 39, number of points 448, running total 7809, all-even 0, all-odd 0, reduced list 2: (1,1,1,6); (1,2,3,5)

distance-squared 40, number of points 144, running total 7953, all-even 144, all-odd 0, reduced list 2: (0,0,2,6); (2,2,4,4)

distance-squared 41, number of points 336, running total 8289, all-even 0, all-odd 0, reduced list 3: (0,0,4,5); (0,1,2,6); (0,3,4,4)

distance-squared 42, number of points 768, running total 9057, all-even 0, all-odd 0, reduced list 4: (0,1,4,5); (1,1,2,6); (1,3,4,4); (2,2,3,5)

distance-squared 43, number of points 352, running total 9409, all-even 0, all-odd 0, reduced list 3: (0,3,3,5); (1,1,4,5); (3,3,3,4)

distance-squared 44, number of points 288, running total 9697, all-even 96, all-odd 192, reduced list 2: (0,2,2,6); (1,3,3,5)

distance-squared 45, number of points 624, running total 10321, all-even 0, all-odd 0, reduced list 4: (0,0,3,6); (0,2,4,5); (1,2,2,6); (2,3,4,4)

distance-squared 46, number of points 576, running total 10897, all-even 0, all-odd 0, reduced list 2: (0,1,3,6); (1,2,4,5)

distance-squared 47, number of points 384, running total 11281, all-even 0, all-odd 0, reduced list 2: (1,1,3,6); (2,3,3,5)

distance-squared 48, number of points 96, running total 11377, all-even 96, all-odd 0, reduced list 2: (0,4,4,4); (2,2,2,6)

distance-squared 49, number of points 456, running total 11833, all-even 0, all-odd 0, reduced list 4: (0,0,0,7); (0,2,3,6); (1,4,4,4); (2,2,4,5)

distance-squared 50, number of points 744, running total 12577, all-even 0, all-odd 0, reduced list 5: (0,0,1,7); (0,0,5,5); (0,3,4,5); (1,2,3,6); (3,3,4,4)

distance-squared 51, number of points 576, running total 13153, all-even 0, all-odd 0, reduced list 3: (0,1,1,7); (0,1,5,5); (1,3,4,5)

distance-squared 52, number of points 336, running total 13489, all-even 112, all-odd 224, reduced list 5: (0,0,4,6); (1,1,1,7); (1,1,5,5); (2,4,4,4); (3,3,3,5)

distance-squared 53, number of points 432, running total 13921, all-even 0, all-odd 0, reduced list 3: (0,0,2,7); (0,1,4,6); (2,2,3,6)

distance-squared 54, number of points 960, running total 14881, all-even 0, all-odd 0, reduced list 5: (0,1,2,7); (0,2,5,5); (0,3,3,6); (1,1,4,6); (2,3,4,5)

distance-squared 55, number of points 576, running total 15457, all-even 0, all-odd 0, reduced list 3: (1,1,2,7); (1,2,5,5); (1,3,3,6)

distance-squared 56, number of points 192, running total 15649, all-even 192, all-odd 0, reduced list 1: (0,2,4,6)

distance-squared 57, number of points 640, running total 16289, all-even 0, all-odd 0, reduced list 4: (0,2,2,7); (0,4,4,5); (1,2,4,6); (3,4,4,4)

distance-squared 58, number of points 720, running total 17009, all-even 0, all-odd 0, reduced list 5: (0,0,3,7); (1,2,2,7); (1,4,4,5); (2,2,5,5); (2,3,3,6)

distance-squared 59, number of points 480, running total 17489, all-even 0, all-odd 0, reduced list 3: (0,1,3,7); (0,3,5,5); (3,3,4,5)

distance-squared 60, number of points 576, running total 18065, all-even 192, all-odd 384, reduced list 3: (1,1,3,7); (1,3,5,5); (2,2,4,6)

distance-squared 61, number of points 496, running total 18561, all-even 0, all-odd 0, reduced list 4: (0,0,5,6); (0,3,4,6); (2,2,2,7); (2,4,4,5)

distance-squared 62, number of points 768, running total 19329, all-even 0, all-odd 0, reduced list 3: (0,1,5,6); (0,2,3,7); (1,3,4,6)

distance-squared 63, number of points 832, running total 20161, all-even 0, all-odd 0, reduced list 4: (1,1,5,6); (1,2,3,7); (2,3,5,5); (3,3,3,6)

distance-squared 64, number of points 24, running total 20185, all-even 24, all-odd 0, reduced list 2: (0,0,0,8); (4,4,4,4)

distance-squared 65, number of points 672, running total 20857, all-even 0, all-odd 0, reduced list 4: (0,0,1,8); (0,0,4,7); (0,2,5,6); (2,3,4,6)

distance-squared 66, number of points 1152, running total 22009, all-even 0, all-odd 0, reduced list 6: (0,1,1,8); (0,1,4,7); (0,4,5,5); (1,2,5,6); (2,2,3,7); (3,4,4,5)

distance-squared 67, number of points 544, running total 22553, all-even 0, all-odd 0, reduced list 4: (0,3,3,7); (1,1,1,8); (1,1,4,7); (1,4,5,5)

distance-squared 68, number of points 432, running total 22985, all-even 144, all-odd 288, reduced list 4: (0,0,2,8); (0,4,4,6); (1,3,3,7); (3,3,5,5)

distance-squared 69, number of points 768, running total 23753, all-even 0, all-odd 0, reduced list 4: (0,1,2,8); (0,2,4,7); (1,4,4,6); (2,2,5,6)

distance-squared 70, number of points 1152, running total 24905, all-even 0, all-odd 0, reduced list 5: (0,3,5,6); (1,1,2,8); (1,2,4,7); (2,4,5,5); (3,3,4,6)

distance-squared 71, number of points 576, running total 25481, all-even 0, all-odd 0, reduced list 2: (1,3,5,6); (2,3,3,7)

distance-squared 72, number of points 312, running total 25793, all-even 312, all-odd 0, reduced list 3: (0,0,6,6); (0,2,2,8); (2,4,4,6)

distance-squared 73, number of points 592, running total 26385, all-even 0, all-odd 0, reduced list 5: (0,0,3,8); (0,1,6,6); (1,2,2,8); (2,2,4,7); (4,4,4,5)

distance-squared 74, number of points 912, running total 27297, all-even 0, all-odd 0, reduced list 5: (0,0,5,7); (0,1,3,8); (0,3,4,7); (1,1,6,6); (2,3,5,6)

distance-squared 75, number of points 992, running total 28289, all-even 0, all-odd 0, reduced list 5: (0,1,5,7); (0,5,5,5); (1,1,3,8); (1,3,4,7); (3,4,5,5)

distance-squared 76, number of points 480, running total 28769, all-even 160, all-odd 320, reduced list 5: (0,2,6,6); (1,1,5,7); (1,5,5,5); (2,2,2,8); (3,3,3,7)

distance-squared 77, number of points 768, running total 29537, all-even 0, all-odd 0, reduced list 4: (0,2,3,8); (0,4,5,6); (1,2,6,6); (3,4,4,6)

distance-squared 78, number of points 1344, running total 30881, all-even 0, all-odd 0, reduced list 4: (0,2,5,7); (1,2,3,8); (1,4,5,6); (2,3,4,7)

distance-squared 79, number of points 640, running total 31521, all-even 0, all-odd 0, reduced list 3: (1,2,5,7); (2,5,5,5); (3,3,5,6)

distance-squared 80, number of points 144, running total 31665, all-even 144, all-odd 0, reduced list 2: (0,0,4,8); (2,2,6,6)

distance-squared 81, number of points 968, running total 32633, all-even 0, all-odd 0, reduced list 6: (0,0,0,9); (0,1,4,8); (0,3,6,6); (0,4,4,7); (2,2,3,8); (2,4,5,6)

distance-squared 82, number of points 1008, running total 33641, all-even 0, all-odd 0, reduced list 7: (0,0,1,9); (0,3,3,8); (1,1,4,8); (1,3,6,6); (1,4,4,7); (2,2,5,7); (4,4,5,5)

distance-squared 83, number of points 672, running total 34313, all-even 0, all-odd 0, reduced list 4: (0,1,1,9); (0,3,5,7); (1,3,3,8); (3,3,4,7)

distance-squared 84, number of points 768, running total 35081, all-even 256, all-odd 512, reduced list 5: (0,2,4,8); (1,1,1,9); (1,3,5,7); (3,5,5,5); (4,4,4,6)

distance-squared 85, number of points 864, running total 35945, all-even 0, all-odd 0, reduced list 5: (0,0,2,9); (0,0,6,7); (1,2,4,8); (2,3,6,6); (2,4,4,7)

distance-squared 86, number of points 1056, running total 37001, all-even 0, all-odd 0, reduced list 5: (0,1,2,9); (0,1,6,7); (0,5,5,6); (2,3,3,8); (3,4,5,6)

distance-squared 87, number of points 960, running total 37961, all-even 0, all-odd 0, reduced list 4: (1,1,2,9); (1,1,6,7); (1,5,5,6); (2,3,5,7)

distance-squared 88, number of points 288, running total 38249, all-even 288, all-odd 0, reduced list 2: (0,4,6,6); (2,2,4,8)

distance-squared 89, number of points 720, running total 38969, all-even 0, all-odd 0, reduced list 5: (0,0,5,8); (0,2,2,9); (0,2,6,7); (0,3,4,8); (1,4,6,6)

distance-squared 90, number of points 1872, running total 40841, all-even 0, all-odd 0, reduced list 9: (0,0,3,9); (0,1,5,8); (0,4,5,7); (1,2,2,9); (1,2,6,7); (1,3,4,8); (2,5,5,6); (3,3,6,6); (3,4,4,7)

distance-squared 91, number of points 896, running total 41737, all-even 0, all-odd 0, reduced list 5: (0,1,3,9); (1,1,5,8); (1,4,5,7); (3,3,3,8); (4,5,5,5)

distance-squared 92, number of points 576, running total 42313, all-even 192, all-odd 384, reduced list 3: (1,1,3,9); (2,4,6,6); (3,3,5,7)

distance-squared 93, number of points 1024, running total 43337, all-even 0, all-odd 0, reduced list 5: (0,2,5,8); (2,2,2,9); (2,2,6,7); (2,3,4,8); (4,4,5,6)

distance-squared 94, number of points 1152, running total 44489, all-even 0, all-odd 0, reduced list 4: (0,2,3,9); (0,3,6,7); (1,2,5,8); (2,4,5,7)

distance-squared 95, number of points 960, running total 45449, all-even 0, all-odd 0, reduced list 3: (1,2,3,9); (1,3,6,7); (3,5,5,6)

distance-squared 96, number of points 96, running total 45545, all-even 96, all-odd 0, reduced list 1: (0,4,4,8)

distance-squared 97, number of points 784, running total 46329, all-even 0, all-odd 0, reduced list 6: (0,0,4,9); (0,5,6,6); (1,4,4,8); (2,2,5,8); (3,4,6,6); (4,4,4,7)

distance-squared 98, number of points 1368, running total 47697, all-even 0, all-odd 0, reduced list 7: (0,0,7,7); (0,1,4,9); (0,3,5,8); (1,5,6,6); (2,2,3,9); (2,3,6,7); (3,3,4,8)

distance-squared 99, number of points 1248, running total 48945, all-even 0, all-odd 0, reduced list 6: (0,1,7,7); (0,3,3,9); (0,5,5,7); (1,1,4,9); (1,3,5,8); (3,4,5,7)

[elpivoxz] cancer causes cancer

cancer metastasis: cancer causes cancri.

inspired by Chantix causing cancer.  (of course, continuing smoking -- not using Chantix -- also causes cancer.)  also inspired by California labeling everything as causing cancer.

how different is the mechanism of cancer metastasis compared to carcinogens causing cancer?  I suspect very different.

[owiwfzux] shuffling one bit

persons A, B, and C in separate rooms with point-to-point communication.

  1. person C describes to person A how they intend to shuffle cards, e.g., 3 riffle shuffles.
  2. person A defines two decks, specifying the order of cards in each deck, and tells the information to person B.
  3. person B secretly randomly chooses one of person A's card orderings, constructs a deck with the chosen ordering, then gives the deck to person C.  we go through this person B intermediary so that person A cannot secretly mark decks to tell them apart afterward.
  4. person C shuffles the received deck as declared in step 1, and gives the shuffled deck to person A.
  5. person A tries to determine which of the two decks person B gave to person C.

I strongly suspect there are ways that person A can construct decks so that it is very difficult to shuffle them enough destroy the one bit of information that is encoded in them.  you will need the full 7 riffle shuffles as proved by Diaconis.  (practically perhaps more, if your riffle shuffles are not very good.)

optionally omit step 1 to give person A less of an advantage.

a "shuffling" step that person C can do to possibly make things difficult is the following: flip a coin.  if and only if tails, reverse the order of the deck.  (I am not aware of any way to do this quickly.  perhaps cardistry magicians know of a way.)  then, do other shuffles.

variation: split the deck into 2.  flip a coin.  reverse one half or the other corresponding to the coin flip.  then riffle shuffle.

also consider person C hand scrambling a speedcube instead of shuffling a deck of cards.  person B needs to be able to construct arbitrary (legal) Rubik's cube states given by person A.

[umgnxrbn] Xen networking on Debian bullseye

mostly following Debian instructions for Xen.

standard stuff added to /etc/network/interfaces:

iface enp58s0f1 inet manual

auto xenbr0
iface xenbr0 inet dhcp
     bridge_ports enp58s0f1

"ip a" does show xenbr0.

(tangent: we would prefer to do "allow-hotplug" instead of "auto" to prevent boot from being slow if there is no ethernet cable plugged in, but allow-hotplug xenbr0 does not work (network does not come up; unsurprisingly udev cannot detect that there's a network cable plugged into "xenbr0").  with "auto", boot is slower, but not the full 5 minutes of delay as the boot message seems to suggest: boot continues after about 1 minute of delay.)

somewhat nonstandard on this system is that NetworkManager is running: NetworkManager manages the wifi interface, and ifupdown (aka /etc/network/interfaces) manages ethernet.  NetworkManager is supposed to avoid trying to manage interfaces in /etc/network/interfaces, as configured in the default /etc/NetworkManager/NetworkManager.conf:

[main]
plugins=ifupdown,keyfile

[ifupdown]
managed=false

however, when bringing up a paraVM created with xen-create-image, we see this in journalctl:

NetworkManager[68473]: <info> [1636093454.5580] settings: (vif7.0): created default wired connection 'Wired connection 1'

and then later more references to vif and Wired connection 1.  for some reason, NetworkManager is, I think, trying to bridge between "Wired connection 1" (which did not even exist!) and the vif for the VM.  the solution is to prevent NetworkManager from touching vif devices:

/etc/NetworkManager/conf.d/no-vif.conf :

[keyfile]
unmanaged-devices=interface-name:vif*

future work: let xenbr0 be a bridge to whichever network connection is active, ethernet or wifi.

[wluwvrvs] old style numerals with cfr-lm on Debian

one of the ways to get old style (lowercase) numerals in LaTeX is with the cfr-lm package.

\usepackage{cfr-lm}

On Debian (buster and bullseye), this package is in the texlive-fonts-extra package.  however, installing the package with --no-install-recommends (or APT::Install-Recommends "false" in /etc/apt/apt.conf.d ) is insufficient; there are required dependencies among the package's Recommends (a bug).

most of the additional needed packages can be tracked down with apt-file, but one error message is not so obvious:

No file OMLlmm.fd.

! LaTeX Error: This NFSS system isn't set up properly.

the missing file is in the lmodern package with a file name without capitalization.  somehow LaTeX knows to search for file names case-insensitively, even though its error messages give file names with case.

lmodern: /usr/share/texmf/tex/latex/lm/omllmm.fd

the complete set of packages needed for cfr-lm is

apt install --no-install-recommends texlive-fonts-extra texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended lmodern

[qlcpuedj] safe primes below many powers of 2

we give the largest safe prime below selected powers of 2.  we also give additional safe primes below them until one which has 2 as a primitive root (generator).  the exponents are rounded decimal powers of 2 (so the sequence is doubly exponential)  (previously, not rounding until the end which requires a large amount of floating point precision.)

the amount of extra work required to find a safe prime with primitive root 2 is Artin's constant.

this work expands on previous work and OEIS A181356.

also previously, safe primes below powers of 10.

for example, in the list below, 2^3.1 ~= 8.57, and round(2^3.1) = 9, so 2^round(2^3.1) = 2^9 = 512.  the largest safe primes less than 512 are 2^9-9 = 512-9 = 503 (which has least primitive root 5), 2^9-33 = 512-33 = 479 (which has least primitive root 13), and 2^9-45 = 512-45 = 467 (which has least primitive root 2), where we stop, having found a safe prime with primitive root 2.  the offsets are -9, -33, and -45.

future post (srzrbviv): source code and implementation discussion.

currently, the recommendation for cryptographic security for Diffie-Hellman key exchange when using the integer discrete logarithm problem is safe prime moduli of size at least 2^11 = 2048 bits.

2^round(2^1.4) = 2^3 : -1 -3
2^round(2^1.5) = 2^3 : -1 -3
2^round(2^1.6) = 2^3 : -1 -3
2^round(2^1.7) = 2^3 : -1 -3
2^round(2^1.8) = 2^3 : -1 -3
2^round(2^1.9) = 2^4 : -5
2^round(2^2.0) = 2^4 : -5
2^round(2^2.1) = 2^4 : -5
2^round(2^2.2) = 2^5 : -9 -21
2^round(2^2.3) = 2^5 : -9 -21
2^round(2^2.4) = 2^5 : -9 -21
2^round(2^2.5) = 2^6 : -5
2^round(2^2.6) = 2^6 : -5
2^round(2^2.7) = 2^6 : -5
2^round(2^2.8) = 2^7 : -21
2^round(2^2.9) = 2^7 : -21
2^round(2^3.0) = 2^8 : -29
2^round(2^3.1) = 2^9 : -9 -33 -45
2^round(2^3.2) = 2^9 : -9 -33 -45
2^round(2^3.3) = 2^10 : -5
2^round(2^3.4) = 2^11 : -9 -21
2^round(2^3.5) = 2^11 : -9 -21
2^round(2^3.6) = 2^12 : -17 -89 -149
2^round(2^3.7) = 2^13 : -45
2^round(2^3.8) = 2^14 : -161 -197
2^round(2^3.9) = 2^15 : -165
2^round(2^4.0) = 2^16 : -269
2^round(2^4.1) = 2^17 : -285
2^round(2^4.2) = 2^18 : -17 -1265 -1661
2^round(2^4.3) = 2^20 : -233 -449 -989
2^round(2^4.4) = 2^21 : -9 -285
2^round(2^4.5) = 2^23 : -321 -729 -1101
2^round(2^4.6) = 2^24 : -317
2^round(2^4.7) = 2^26 : -677
2^round(2^4.8) = 2^28 : -437
2^round(2^4.9) = 2^30 : -1385 -1697 -2645
2^round(2^5.0) = 2^32 : -209 -1409 -3509
2^round(2^5.1) = 2^34 : -641 -1001 -1637
2^round(2^5.2) = 2^37 : -45
2^round(2^5.3) = 2^39 : -381
2^round(2^5.4) = 2^42 : -2201 -3737 -5417 -12581
2^round(2^5.5) = 2^45 : -573
2^round(2^5.6) = 2^49 : -2709
2^round(2^5.7) = 2^52 : -473 -2729 -2933
2^round(2^5.8) = 2^56 : -2249 -2837
2^round(2^5.9) = 2^60 : -3677
2^round(2^6.0) = 2^64 : -1469
2^round(2^6.1) = 2^69 : -165
2^round(2^6.2) = 2^74 : -545 -9521 -22745 -23777 -26045
2^round(2^6.3) = 2^79 : -2001 -2709
2^round(2^6.4) = 2^84 : -5297 -7013
2^round(2^6.5) = 2^91 : -81 -5661
2^round(2^6.6) = 2^97 : -6909
2^round(2^6.7) = 2^104 : -15773
2^round(2^6.8) = 2^111 : -429
2^round(2^6.9) = 2^119 : -3981
2^round(2^7.0) = 2^128 : -15449 -21509
2^round(2^7.1) = 2^137 : -849 -1785 -2289 -30189
2^round(2^7.2) = 2^147 : -2601 -15201 -30249 -38145 -44841 -49761 -92565
2^round(2^7.3) = 2^158 : -665 -14117
2^round(2^7.4) = 2^169 : -20493
2^round(2^7.5) = 2^181 : -20265 -83793 -96093
2^round(2^7.6) = 2^194 : -6641 -37961 -38057 -42257 -74681 -86801 -139565
2^round(2^7.7) = 2^208 : -4973
2^round(2^7.8) = 2^223 : -28929 -44901
2^round(2^7.9) = 2^239 : -87429
2^round(2^8.0) = 2^256 : -36113 -188069
2^round(2^8.1) = 2^274 : -54605
2^round(2^8.2) = 2^294 : -184181
2^round(2^8.3) = 2^315 : -51321 -61245
2^round(2^8.4) = 2^338 : -35861
2^round(2^8.5) = 2^362 : -169805
2^round(2^8.6) = 2^388 : -17297 -156377 -222113 -365369 -375737 -497693
2^round(2^8.7) = 2^416 : -222749
2^round(2^8.8) = 2^446 : -324857 -835781
2^round(2^8.9) = 2^478 : -699821
2^round(2^9.0) = 2^512 : -38117
2^round(2^9.1) = 2^549 : -1005069
2^round(2^9.2) = 2^588 : -313793 -406997
2^round(2^9.3) = 2^630 : -104585 -636521 -813917
2^round(2^9.4) = 2^676 : -322229
2^round(2^9.5) = 2^724 : -52517
2^round(2^9.6) = 2^776 : -356033 -1710317
2^round(2^9.7) = 2^832 : -1281293
2^round(2^9.8) = 2^891 : -1926369 -2217081 -4672149
2^round(2^9.9) = 2^955 : -1077885
2^round(2^10.0) = 2^1024 : -1093337 -1370753 -1428353 -1503509
2^round(2^10.1) = 2^1097 : -3161265 -3560373
2^round(2^10.2) = 2^1176 : -2056193 -2069357
2^round(2^10.3) = 2^1261 : -2634393 -4262745 -4668993 -4944945 -5565909
2^round(2^10.4) = 2^1351 : -25905 -1138665 -3569025 -3720261
2^round(2^10.5) = 2^1448 : -549269
2^round(2^10.6) = 2^1552 : -551729 -6104357
2^round(2^10.7) = 2^1663 : -611625 -774609 -1455849 -1719765
2^round(2^10.8) = 2^1783 : -1233705 -3645345 -4163709
2^round(2^10.9) = 2^1911 : -3231309
2^round(2^11.0) = 2^2048 : -1942289 -4801589
2^round(2^11.1) = 2^2195 : -3502989
2^round(2^11.2) = 2^2353 : -1678485
2^round(2^11.3) = 2^2521 : -4968453
2^round(2^11.4) = 2^2702 : -246797
2^round(2^11.5) = 2^2896 : -761717
2^round(2^11.6) = 2^3104 : -4210193 -8338889 -10214933
2^round(2^11.7) = 2^3327 : -775149
2^round(2^11.8) = 2^3566 : -16802105 -37290401 -56582861
2^round(2^11.9) = 2^3822 : -9543617 -11776781
2^round(2^12.0) = 2^4096 : -10895177 -13463237
2^round(2^12.1) = 2^4390 : -77336201 -79007045
2^round(2^12.2) = 2^4705 : -24442605
2^round(2^12.3) = 2^5043 : -1559505 -12588225 -40350225 -44478165
2^round(2^12.4) = 2^5405 : -81935433 -86932389
2^round(2^12.5) = 2^5793 : -11998425 -33851673 -36961425 -50712993 -83738889 -236665449 -254515869
2^round(2^12.6) = 2^6208 : -20057957
2^round(2^12.7) = 2^6654 : -11162645
2^round(2^12.8) = 2^7132 : -7052309
2^round(2^12.9) = 2^7643 : -44120649 -123495021
2^round(2^13.0) = 2^8192 : -43644929 -49930517
2^round(2^13.1) = 2^8780 : -22581833 -25207709
2^round(2^13.2) = 2^9410 : -10428521 -103539077
2^round(2^13.3) = 2^10086 : -29814917
2^round(2^13.4) = 2^10809 : -6562329 -216052833 -262814913 -415344465 -429846909
2^round(2^13.5) = 2^11585 : -370812549
2^round(2^13.6) = 2^12417 : -103708449 -294381693
2^round(2^13.7) = 2^13308 : -10956077
2^round(2^13.8) = 2^14263 : -397079661
2^round(2^13.9) = 2^15287 : -214216065 -312159765
2^round(2^14.0) = 2^16384 : -364486013
2^round(2^14.1) = 2^17560 : -141642533
2^round(2^14.2) = 2^18820 : -339964313 -571288529 -898757417 -1027256537 -1156399229
2^round(2^14.3) = 2^20171 : -276625749
2^round(2^14.4) = 2^21619 : -674266449 -833225181
2^round(2^14.5) = 2^23170 : -259994441 -366289061
2^round(2^14.6) = 2^24834 : -415810805
2^round(2^14.7) = 2^26616 : -2909149877
2^round(2^14.8) = 2^28526 : -520866521 -811123361 -990961337 -1485278921 -1487483597
2^round(2^14.9) = 2^30574 : -568083185 -1447030925
2^round(2^15.0) = 2^32768 : -718982153 -2543122349
2^round(2^15.1) = 2^35120 : -881530949

[dclecbyo] 20 centers of projection

assume a map projection in which distortion is small near the center of projection and large toward the edges, e.g., orthographic.  depict a sphere as a collection of 12 or 20 maps centered at the vertices of an icosahedron or dodecahedron respectively.  the maps will overlap, perhaps by a lot.

12 = 3*4 and 20 = 4*5, so both numbers are good for putting the maps in an elegant rectangular array.  if one avoids placing centers of projection at the poles, north can always be up.

[bhxtejtu] x and 1-x proportions

order individuals by wealth and compute the running sum: Lorenz curve.  there is a special proportion p such that the top p of the population holds 1-p of the wealth, e.g., the top 1% holds 99% of the wealth.

hypothesize that smaller p means greater inequality.  another similar measure is the Gini coefficient.

is it guaranteed that there is only one special p?

can p be read off a Lorenz curve?

what if the poorest have negative wealth?

[tkhqrixn] hash of primes

print primes in decimal, one per line.  partition into intervals from 2^n to 2^(n+1)-1.  hash each partition with MD5.

goal is to verify whether a prime number generator is working properly.  here are results from primesieve version 6.3 packaged on Ubuntu 18.04 .  future work: repeat with some other way of generating primes.

the first result is the MD5 hash of the empty string:

primesieve -p 1 1 | md5sum
d41d8cd98f00b204e9800998ecf8427e -

the next result is the hash of the string "2\n3\n":

primesieve -p 2 3 | md5sum
19283599a9866154a20cbb0be6adc1bc -

the next result is the hash of the string "5\n7\n":

primesieve -p 4 7 | md5sum
ec4aab475ce80bfd5469640c71b17108 -

primesieve -p 8 15 | md5sum
ea40c553d35c3769e4563356e5571312 -

primesieve -p 16 31 | md5sum
389ee0bc459bb738682302eaeeffdb9e -

primesieve -p 32 63 | md5sum
4625d9d99b82ddd64b2a7f744e81b5a2 -

primesieve -p 64 127 | md5sum
0d59f9c405e9f4ddd6a9dff34d995747 -

primesieve -p 128 255 | md5sum
24ac923d7d3fcb9ee30f660db48a896a -

primesieve -p 256 511 | md5sum
0def87cc99e15b8537b4fa735e9e173e -

primesieve -p 512 1023 | md5sum
09f3b02a4e83b28eb353da6b143173e5 -

primesieve -p 1024 2047 | md5sum
9a0e27c6dd0eb9c6f16896b1a19504f9 -

primesieve -p 2048 4095 | md5sum
f8812a2f97f570cc893c089aafacf7df -

primesieve -p 4096 8191 | md5sum
4dabac6164589732ef6f9ffa843c1125 -

primesieve -p 8192 16383 | md5sum
62b0b80451cc9e02de892c93cd5cac34 -

primesieve -p 16384 32767 | md5sum
4969d628ab6f72295afa9e663542cb59 -

primesieve -p 32768 65535 | md5sum
a90e03d5ce780a57bd962016af6a88b9 -

primesieve -p 65536 131071 | md5sum
7e9e7de2250a57a7aab73d8a347234c0 -

primesieve -p 131072 262143 | md5sum
725762c6e5d246e32a71fff4490eb345 -

primesieve -p 262144 524287 | md5sum
b87228cac6ba574ce059b06a89cf743b -

primesieve -p 524288 1048575 | md5sum
9db8a44d1018406bed9ddd810fdf7ed7 -

primesieve -p 1048576 2097151 | md5sum
00220343780b1453c9b7a634c4bcf395 -

primesieve -p 2097152 4194303 | md5sum
0b007f8da982f6bb8405bc1a987479ab -

primesieve -p 4194304 8388607 | md5sum
fbf768c73a8b3808ad987616df137937 -

primesieve -p 8388608 16777215 | md5sum
cbcd6f4bba8e755e38fec96f700bc6e4 -

primesieve -p 16777216 33554431 | md5sum
6be429185d33bf792dec21a9e2f838a8 -

primesieve -p 33554432 67108863 | md5sum
ba6d3bfb1b1fd482c2f6c2068e6def69 -

primesieve -p 67108864 134217727 | md5sum
5690fcae89b922e83149cde919d13b08 -

primesieve -p 134217728 268435455 | md5sum
e01b4b2f2665faa58a6c7c0e570c5623 -

primesieve -p 268435456 536870911 | md5sum
5e73b7013ea260fbc89689d196867c7f -

primesieve -p 536870912 1073741823 | md5sum
baacef4b4e06204e8c70e9e7b70caf51 -

primesieve -p 1073741824 2147483647 | md5sum
72922f6383a3a6003c3a7d648bb8e3da -

primesieve -p 2147483648 4294967295 | md5sum
9c1e25fef3b6abc9a1dca782bf5d422c -

primesieve -p 4294967296 8589934591 | md5sum
b45d191f562952acb0fe37aff3d92733 -

primesieve -p 8589934592 17179869183 | md5sum
e1ab401038abbef1283e57ca6f058e4b -

primesieve -p 17179869184 34359738367 | md5sum
153ae06a21b0fedce43b4fcd43c5ff6b -

primesieve -p 34359738368 68719476735 | md5sum
b0c7f4f0c04ffca5c3bb22bc48a8e5e1 -

primesieve -p 68719476736 137438953471 | md5sum
d0f12246675c9b5c9c3a373a509d42a1 -

primesieve -p 137438953472 274877906943 | md5sum
64e90d67029fc0e6deed56a0f564aad4 -

primesieve -p 274877906944 549755813887 | md5sum
26b043e2db96fb2c5e80da3898507372 -

primesieve -p 549755813888 1099511627775 | md5sum
6b7104f5a6e8dcac949350888a9df58a -

primesieve -p 1099511627776 2199023255551 | md5sum
71ee6df793118e85829074b152e59c73 -

primesieve -p 2199023255552 4398046511103 | md5sum
c420df38251cada4f31fb9df35dde6f1 -

primesieve -p 4398046511104 8796093022207 | md5sum
95c931c24f7a99483ac6d16b5530d838 -

primesieve -p 8796093022208 17592186044415 | md5sum
157266729337657753d52198f32a2698 -

primesieve -p 17592186044416 35184372088831 | md5sum
5d98aa14014d80dc646409f9103c9949 -

primesieve -p 35184372088832 70368744177663 | md5sum
6aa4896128011565848ccfad6f00418b -

the final result above is the MD5 checksum of all 46-bit primes that have their most significant bit set, that is, primes between 2^45 = 35184372088832 and 2^46 - 1 = 70368744177663 .

future work: this seems amenable to parallelization and tree hashing.  let the leaves of the tree be the bitstring of prime and composite.  reuse previously hashed results, especially of long strings of consecutive composites.

[tfzwneso] cave system outside non-overlapping cubes

start with some random points.  grow circles around them, all circles growing at the same rate.  a circle stops growing when it touches another circle.  this results in a circle packing.  (not a very tight circle packing: what is a better way to generate denser random circle packings?  maybe Apollonian gasket.)

same thing but with axis-aligned growing squares instead of circles.  what is the nature of the space outside of all squares after they have grown as much as possible?  is it typically connected?

similar thing with rectangles.  when a side touches another side, growth stops only in that direction.  the other 3 sides keep growing until they too are each impeded.  there needs to be an exterior bounding box.  vaguely has a similar feeling to Voronoi partition.  with this much additional freedom to grow compared to squares, there probably will not be much remaining space, and the remaining space probably won't be very connected.

same thing except with bricks (rectangular parallelepipeds) in 3D.  3D probably has enough space for the gaps between bricks to typically remain connected.

motivation is to generate a cave system for a game, a collection of rooms connected by twisty passages.  with bricks, everything axis-aligned is like Minecraft.  we want rooms and passages to have distinctive shapes.  probably best if character can climb everywhere, including ceiling.

define a room to be a gap large enough to fit a cube of given size.  (or sphere?)  what is the extent of a room?  define a center as a point surrounded by an empty cube of the given size, then define a room from the connected component of centers.  devilish detail remains of choosing the size of the probe cube.

bricks could grow at different rates, or not all start growing at the same time.  inspired by crystal growth in the Cave of the Crystals.  the cave is the free space not occupied by crystals.

"epoxy" to fill in gaps.  tunneling to make more passages.

[mcbkqach] those who will never be vaccinated

  1. if I am an illegal immigrant and choose to be vaccinated against COVID-19, might the information I provide on vaccination paperwork be used to discover my illegality and hasten my deportation?

  2. if I am evading law enforcement, perhaps for a crime that hasn't been discovered yet, might the information I provide or leak during vaccination be used to apprehend me?  along with information directly provided on vaccination paperwork, leaked information might include becoming filmed on security cameras or automated license-plate recording cameras, or being tracked by cell phone tracking systems.

these are rhetorical questions; the answers to both is "obviously yes".  we do not have any mechanisms to credibly convince people of these groups that the answer is "no".  law enforcement, including immigration, cannot be stopped in accessing (via subpoena) any of the above information.

included in group 2 are people who are evading paying child support.

also included in group 2 might be a very large number of people trying to keep a low profile because of "three felonies a day".

also included in group 2 might be people who self-recognize that they have a predilection toward "antisocial behavior", and might do something criminally antisocial, or might do something antisocial that attracts attention of law enforcement leading to arrest for a different crime, during the course of being vaccinated.  antisocial behavior, as judged by our racist society, might include "being black".

what is the total population of the people in these categories?  assuming they rationally choose not be vaccinated because they don't want to be deported or end up in jail or have wealth confiscated, are they alone enough to prevent general herd immunity?  people in these groups likely geographically cluster.  regions with high concentrations of such people are even less likely to locally reach herd immunity.

we have created a society with a large number of such people.  benefits perhaps have included cheap labor, law and order.  but we now pay its cost -- pay the piper -- in situations like this, being unable to stop a disease -- those who are not vaccinated will breed new COVID-19 variants -- and being unable to return to normalcy.

it seems it will require a mind-bogglingly vast restructuring of society to significantly decrease the population of such people.

[bekvhdwz] ECC signature QR code

elliptic curve cryptography (ECC) offers signatures considerably shorter than RSA or integer Diffie-Hellman.  we examine the signature size of Ed25519.  (constructing an Ed25519 collision requires 2^128 work.)

an Ed25519 signature is 64 bytes, 512 bits, or 155 digits.  this fits within a version 4 (size) QR code at ECC (error correcting code, an unfortunate collision of acronyms) level L, or version 8 at level H.  note: the QR code examples below do not encode real signatures or public keys.  instead, they merely encode digit strings with the right lengths (future post duartbli).  although QR codes can encode raw binary data, we've chosen base-10 digit strings with the same amount of information because many QR code readers don't do so well with binary data.  (imagine that radix conversion has been done.)

QR code 155 digits, level L   QR code 155 digits, level H

it's helpful to identify the public key corresponding to the signature.  rather than a key identifier (as usually done with RSA or DSA, discussed below), public keys for ECC are short enough to be included in their entirety.  Ed25519 public keys are 32 bytes, 256 bits, or 78 digits.  here are QR codes encoding a string consisting of "ED25519:", 78 digits (representing a public key), a separator ":", and 155 digits (representing a signature).  it is version 6 at ECC L, version 10 at H.  (we use colon as a separator for maximum efficiency, though it turns out not to matter.)

QR code 71 digits :ED25519: 155 digits, level L   QR code 71 digits :ED25519: 155 digits, level H

note that the payload, i.e., the message which was signed, is not included in the above examples.  given the payload and the public key, the signature can be verified in place; no additional information is required.

the problem of establishing that an included public key can be trusted is the giant open problem of PKI.  (additional information is required.)

here is a nice illustration of what Ed25519 private and public keys and signatures contain: https://blog.mozilla.org/warner/2011/11/29/ed25519-keys/

for reference, here are possible sizes of public key identifiers in PGP (gnupg, gpg, openpgp).  all of these could be applied to Ed25519 but would require standardization.

32-bit key ID (4 bytes, 10 digits)
64-bit key ID (8 bytes, 20 digits)
MD5 key fingerprint (128 bits, 16 bytes, 39 digits)
SHA-1 key fingerprint (160 bits, 20 bytes, 49 digits)

[avmeugtp] it's raining snakes

as might be expected for someone who has "had enough", Samuel L. Jackson snaps, suicidally doing something that causes mid-air destruction of the aircraft.  perhaps using a  purple lightsaber.

down below, as humor or horror, the running gag is snakes falling from above when you least expect it.

snakes from a plane.

previously: (2), (3).

[mcrpntzv] downfall

entitle the famous scene of Yitzhak Rabin and Yassar Arafat shaking hands, with Bill Clinton behind them, "Downfall" (13 September 1993, Oslo Accords).

Rabin would go on to be assassinated by an Israeli citizen unhappy with Israel making peace with the Palestinians.  Israelis generally agreed with the assassination, choosing to replace Rabin (actually Rabin's successor) with a hard-line government bent on making war with Palestine.

Arafat's PLO would go on go be supplanted by Hamas, a hard-line government bent on making war with Israel.  along the way, Arafat dies, possibly assassinated by that Isareli government bent on making war with Palestine.

Clinton's political party would quickly go on to lose both houses of Congress (Contract with America), and later the Presidency, to a hard-line party bent on antisemitism and Islamophobia.

[hzwtbcrh] slow-growing integer exponentials

the sequence a[n] = 2^n, the powers of 2, has growth rate 2.

a[n] = a[n-1] + a[n-2], Fibonacci sequence (also Lucas sequence), powers of the matrix [1 1 ; 1 0], growth rate = golden ratio = 1.618 = Roots[1+x-x^2==0,x] = (1+sqrt(5))/2.

growth rate is the largest eigenvalue of the matrix.  golden ratio seems to be the slowest constant growth possible with 2x2 integer matrix.  (but doubling every two iterations, average growth sqrt(2) = 1.4 is also possible.)

longer lag a[n] = a[n-1] + a[n-3], OEIS A000930 (also A179070), powers of matrix [1 0 1; 1 0 0 ; 0 1 0], growth rate = supergolden ratio = 1.46557123187676802665673122522 = Roots[1+x^2-x^3==0,x] = (1 + (29/2 - (3*sqrt(93))/2)^(1/3) + ((29 + 3*sqrt(93))/2)^(1/3))/3

longer lag a[n] = a[n-1] + a[n-4], powers of matrix [1 0 0 1; 1 0 0 0 ; 0 1 0 0 ; 0 0 1 0], growth rate 1.38027756909761411567330169182 = Roots[1+x^3-x^4==0,x] = 1/4 + sqrt(1 - 16*(2/(3*(-9 + sqrt(849))))^(1/3) + 2*(2/3)^(2/3)*(-9 + sqrt(849))^(1/3))/4 + sqrt(1/2 + 4*(2/(3*(-9 + sqrt(849))))^(1/3) - ((-9 + sqrt(849))/2)^(1/3)/3^(2/3) + 1/(2*sqrt(1 - 16*(2/(3*(-9 + sqrt(849))))^(1/3) + 2*(2/3)^(2/3)* (-9 + sqrt(849))^(1/3))))/2

the quintic case surprisingly has a closed-form growth rate: a[n] = a[n-1] + a[n-5], powers of matrix [1 0 0 0 1; 1 0 0 0 0; 0 1 0 0 0; 0 0 1 0 0 ; 0 0 0 1 0], growth rate 1.3247179572447460260 = Roots[1+x^4-x^5==0,x] = (27/2 - (3*sqrt(69))/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3).  the polynomial factors 1 + x^4 - x^5 = (1 - x + x^2) * (1 + x - x^3) .  the real root (that of the cubic factor), the growth rate, is the plastic number: see Padovan sequence below.

sextic: a[n] = a[n-1] + a[n-6], growth rate 1.28519903324534936790726046413 = N[Roots[1+x^5-x^6==0,x],30]

the next polynomial that factors is 1 + x^10 - x^11 = (1 - x + x^2) * (1 + x - x^3 - x^4 + x^6 + x^7 - x^9) .  the quadratic factor is the same as in the quintic case.

what is the growth rate as function of lag?

because these sequences are powers of a matrix, one can compute high terms very quickly using exponentiation by squaring.  the matrices contain repeated values.  how can this be exploited to compute quicker?

if we take the two oldest numbers, growth rate is slower:

Padovan sequence OEIS A000931 (also Perrin sequence A001608): a[n] = a[n-2] + a[n-3], powers of matrix [0 1 1; 1 0 0 ; 0 1 0], growth rate = plastic number = 1.32471795724474602596090885448 = Roots[0==1+x-x^3,x] = (27/2 - (3*sqrt(69))/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3)

a[n] = a[n-3] + a[n-4], OEIS A017817, growth rate = 1.22074408460575947536168534911 = Roots[0==1+x-x^4,x] = sqrt(-4*(2/(3*(9 + sqrt(849))))^(1/3) + ((9 + sqrt(849))/2)^(1/3)/3^(2/3))/2 + sqrt(4*(2/(3*(9 + sqrt(849))))^(1/3) - ((9 + sqrt(849))/2)^(1/3)/3^(2/3) + 2/sqrt(-4*(2/(3*(9 + sqrt(849))))^(1/3) + ((9 + sqrt(849))/2)^(1/3)/3^(2/3)))/2

quintic case a[n] = a[n-4] + a[n-5], growth rate = 1.16730397826141868425604589985 = Roots[0==1+x-x^5,x].

growth rate seems to be 1+ln(2)/lag for large lag (via Mathematica and Inverse Symbolic Calculator).  more series terms?

previously, evaluating recursive sequences modulo N.

motivation was to test something on integer inputs varying in scale exponentially, but the inputs should not have easy factorizations.  a first attempt, the sequence interleaving 2^n and 3*2^n grows more slowly than 2^n but has all easy factorizations.  Fibonacci is better (though it does have algebraic factorizations) but has no degrees of freedom to control growth rate.  a^n+1 and a^n-1 for integer a and n have algebraic factorizations: Cunningham project.  rounded decimal powers of two.

[dzyomchb] symmetrically distributing points on a hypersphere

the vertices of a geodesic polyhedron distribute points evenly and symmetrically around a sphere (2-sphere, embedded in 3 dimensions).  it is pretty.

is there an analogous family of pretty distributions of points on the 3-sphere, embedded in 4 dimensions?

when there are many vertices on a 2-sphere, the vertex figure around most vertices is a hexagon, reflecting the optimality of hexagonal close packing of discs on a plane.

on the 3-sphere, we analogously expect most vertices to be packed as if at the centers of spheres in a fcc or hcp close packing.  assuming fcc because it has more symmetry, most Voronoi cells around vertices will be approximate rhombic dodecahedra.

on the 2-sphere, for any geodesic polyhedron based on the icosahedron, there are 12 special vertices surrounded by only 5 triangles, not 6.  the Voronoi cell around such a vertex is a pentagon.

what is the maximum symmetry the analogous 4D geodesic polytope can have?  is it the symmetry of the 600-cell?  or the tesseract?  what polyhedron is the Voronoi cell around weird vertices, and how does that polyhedron connect to the rhombic dodecahedra that make up most of the cells?  it probably needs quadrilateral faces.

maybe it has to be the symmetry of the tesseract, and the weird vertices are surrounded by cubes, and the rhombic faces of the rhombic dodecahedra get distorted into squares to meet the faces of the cubes.  all this is wild speculation.

what happens on 4-spheres (in 5D) and beyond?  we remain interested only in pretty distributions of points.  high density lattice packings of identical hyperspheres in Euclidean spaces of various (low) dimensions are known.  for large numbers of points on an N-sphere, distortions of these sphere packings will form "most" of the points on the hypersphere.

in higher dimensions, regular polytopes become boring: only the symmetry of the hypercube is available.  but maybe asking that the symmetry of a point distribution have the symmetry of a regular polytope asks too much.  for example, the E8 polytope and its associated symmetry are not regular.

[hrocucwq] hyperaccelerated bongcloud proof game

the main line way to play the Hyperaccelerated Bongcloud chess opening is

1. Ke1xe2

however, those who are sticklers about "rules" may prefer reaching the position via transposition, for example:

1. Nf3 Nc6   2. Ng1 Nd4   3. Nf3 Nxe2   4. Ng1 Nd4   5. Nf3 Nc6   6. Ke2 Nb8   7. Ng1

[tpywqclf] mazes with doors

let buttons in the maze control internal doors, closing or opening them.  this is a maze with state.

straightforward: only 1 button reachable initially.  pushing it opens a (probably distant) door making the next button accessible, and so forth.  pushing a "used" button again has no effect.  the last button opens a door making the exit accessible.  design the locations of the buttons so that one necessarily visits a large portion of the maze.

variations:

all buttons always accessible.  maze is always solvable no matter what the button state.  implement this by having the combined button state be the seed for a random (solvable) maze generator: pushing any button (probably) radically changes the maze.  perhaps goal is shortest solution.  or shortest solution that turns on all buttons.

a way to generate that doesn't radically change the maze for each button press: partially generate the maze with a fixed PRNG, then finish it off with a new PRNG seeded by button state.  how well this works depends heavily on the chosen maze-generating algorithm.  "doesn't radically change the maze" is subjective.

count the number of "on" buttons.  let this number be the seed for the maze generator.  it feels a little bit like a 3D maze with weird elevators: an elevator only goes one direction and a distance of one floor no matter what floor, but after being used, only goes the other direction.  maybe a weird stairwell or ladder.

partition the buttons into two sets (perhaps visually).  the tuple of the count of on buttons for each set is the seed for a maze generator.  one wanders a "meta" plane of mazes.  perhaps this meta plane is itself a maze with obstacles, so sometimes buttons cannot be pushed even if you reach them.

buttons are not optional.  being in a cell with a button automatically pushes or unpushes it.  it's more like a motion sensor than a button.  or, movement between two adjacent cells triggers the change in doors of the maze.  or, movement in a particular direction.  all of these result in mazes in which you can't easily go backward, though a UI could provide undo.

pushing a button flips the state (xor) of a set of doors associated with the button.

some of these will require effort to ensure that the maze is solvable.

[honhnynk] endgames on infinite boards

investigate chess endgames on infinite boards (including half infinite, quarter infinite, 1/8 infinite) by investigating them on sufficiently large but finite boards.  if the defending king makes it to certain edges, it is deemed to have a strategy that escapes to infinity so the game is drawn.

because everything is finite, tablebases are possible.

these finite regions only approximate infinity, but perhaps a sufficiently large regions can approximate infinity close enough for the endgame evaluation to be exact.

this probably has a chance of working only if the defending side does not have ranged pieces (bishop, rook, queen).

many tricky details remain.  might not work at all.

vaguely similar to angel versus square eater.

this will not work on a chess variant in which perpetual check wins.

previously, endgames on arbitrary but finite boards.

[lhcbianh] twice fun

having fun is inherently enjoyable.  then, having fun, especially in circumstances with adversity, also signals good mental health, high self-qi.

maybe having fun isn't ever inherently enjoyable: maybe it's this signaling that's only ever being enjoyed.  would it be as much fun if you were alone and couldn't tell anyone?

what things require adversity to be fun?  it wouldn't be as fun if the adversity were removed.

possible examples: scary amusement park rides, horror movies.

hazing.

previously similar: validation is fun.

future post: sex is fun.

Thursday, December 02, 2021

[abjimsnm] gaze into the abyss

some German spelling variations in a famous quote from Beyond Good and Evil (Jenseits von Gut und Böse) (1886) by Friedrich Nietzsche:

Wer mit Ungeheuern kämpft, mag zusehn/zusehen, dass/daß er nicht dabei zum Ungeheuer wird.  Und wenn du lange in einen Abgrund blickst, blickt der Abgrund auch in dich hinein.

dass/daß is eszett (sharp s).  daß is probably more correct, but dass seems more common in collections of quotes.  what was German orthography in 1886?

zusehen seems to be more correct, but zusehn seems to be a valid archaic spelling.

the quote comes from a chapter of sayings and provides no further context.  in reality, becoming a monster is complicated (investigated by Zimbardo and many others), and probably requires much, much more than just gazing into the abyss.  though perhaps Zimbardo gazed into the abyss too long himself, becoming evil.

I like the alternate translation "stare into the void; the void stares back", though that changes the meaning.  "abyss" implies looking downward, toward evil, whereas "void" makes no ethical or moral judgment.  and I like the weirdness of the void (nothingness) staring back, in contrast to the (implied) monsters of the abyss staring back.  does spending mental effort on a subject seemingly devoid of ethical and moral issues (e.g., mathematics) turn you evil?

Sunday, November 28, 2021

[znqttiii] non-polynomials

here are some subjectively interesting functions that are not polynomials, accompanied with a range over which the function does not do anything weird (no singularities).

motivation was to test numerical integration.  many quadrature schemes are exact for low-degree polynomials (future post), so we avoid polynomials.

we give the functions in Mathematica syntax for convenience of integration, followed by notes.  some integrals Mathematica cannot do.  all of these can be evaluated quickly to high precision by some way other than numerical integration, sometimes a special function, given in notes.

do some quadrature schemes work poorly if the first derivative is infinite (i.e., function is vertical) at an endpoint?

Integrate[Exp[-x^2/2]/Sqrt[2*Pi],{x,0,2}] : standard normal distribution, Erf, inflection at 1

Integrate[Sin[x],{x,0,Pi}] : integrating sine over a much larger range would even more strongly avoid polynomiality

Integrate[Cos[x],{x,0,Pi/2}]

Integrate[Cos[n*tau-x*Sin[tau]]/Pi,{tau,0,Pi}] : BesselJ[n,x], limits fixed, let n=1 x=10

Integrate[x^x,{x,0,1}] : sophomore's dream, limits fixed, defined as limit at 0, vertical at 0

Integrate[x^-x,{x,0,1}] : sophomore's dream, limits fixed, defined as limit at 0, vertical at 0

Integrate[Sqrt[x],{x,0,1}] : vertical at 0

Integrate[x^((1+Sqrt[5])/2-1),{x,0,1}] : vertical at 0, exponent is golden ratio

Integrate[Log[x],{x,1,E}]

Integrate[1/x,{x,1,E}]

Integrate[Exp[x],{x,0,1}]

Integrate[Exp[x]/x,{x,1/4,2}] : exponential integral

Integrate[Exp[-1/x],{x,0,1}] : exponential integral, defined as limit at 0, flat at zero

Integrate[1/Log[x],{x,0,1/2}] : logarithmic integral, defined as limit at 0

Integrate[Sqrt[1-x^2],{x,-1,1}] : semicircle area, vertical at endpoints

Integrate[Sqrt[1/(1-x^2)],{x,-1/2,1/2}] : 1/6 circle perimeter, we avoid integrating from -1 and 1 because function is undefined at endpoints

Integrate[1/(1+x^2),{x,0,1}] : atan(1) = pi/4

Integrate[Sqrt[1-k^2*Sin[x]^2], {x,0,Pi/2}] : complete elliptic integral of the second kind, arithmetic-geometric mean, limits fixed, ellipse perimeter, let k=255/256, we have used trigonometric form to avoid the function being undefined at endpoints

Integrate[1/Sqrt[1-k^2*Sin[x]^2], {x,0,Pi/2}] : complete elliptic integral of the first kind, limits fixed, let k=15/16, arithmetic-geometric mean, we have used trigonometric form to avoid the function being undefined at endpoints

Integrate[x^a*(1-x)^b, {x,0,1}] : beta function, limits fixed, let a=1/3 b=1/3, vertical at endpoints

future work: rescale, compute Taylor expansions around the midpoint of the interval.

Saturday, November 20, 2021

[nbejauna] Lovers' Waltz cadenza

The Lover's Waltz by Jay Ungar and Molly Mason has a brief cadenza before its final note.  could the cadenza be expanded?

dancing to a non-rhythmic cadenza requires significant departure from basic waltz.

[xiyniyeo] Ocean's 8 acrobat

in Ocean's 8, Qin Shaobo reprises his role from previous films of the franchise.  but in having him, the producers missed a great opportunity to keep it an "all-female crew": there are a great many female gymnasts, acrobats, and contortionists who could have performed that (small) role, probably more available than male.  some of them (e.g., Olympic gymnasts, Youtube stars) are even celebrities, if they wanted to follow the movie's theme of celebrity cameos all over the place.

[dndvumus] most annoying country to map

given a connected region on a sphere, project it onto a plane.  some shapes of regions necessarily have more distortion than others, no matter what map projection is used.  ("amount of distortion" needs to be precisely defined.)  which shapes?

it's not strictly area: a long thin band almost circling a great circle can be flattened into a rectangle without much distortion.  or a star-shaped union of such bands.  we only care about the internal distortion within a shape after projection, and not, say, that the distance between consecutive star endpoints is wrong.

perhaps the badness of a shape is proportional to the size of its largest inscribed circle (spherical cap).

consider only regions without internal holes. (merge the internal hole into the region if necessary.)  map non-contiguous regions by treating each connected component separately (as done with Alaska and Hawaii on typical U.S. maps.)

[lrfhagot] alcohol and evolution of mental health

beer is proof that God wants us to self-medicate for mental health problems.

is it God's love, or evolution?  that is, are the evolution of mental health diseases and the development of alcohol as a technology of human civilization linked?  perhaps mental health issues that could not be treated (somewhat) with alcohol were evolutionarily bred out.

other recreational drugs: have any been cultivated (genetically engineered) for as long as alcohol has been brewed?  long enough to affect human evolution?

Friday, November 19, 2021

[vonobuym] astronomical dust to dust

in astronomy, dust is interstellar material larger than a molecule.  how large is a dust particle permitted to be?

a human corpse floating in space would probably be considered by astronomers as dust.  the proverb "dust to dust" is literally true as scientific jargon.

dust absorbs radiation and reemits with a black body spectrum.  a living human body maintains a constant body temperature (homeostasis), and plants do photosynthesis, so, in general, living things are more complicated than dust in terms of radiation.

can planets and asteroids be dust?  geologically active planets emit radiation unrelated to insolation, and atmospheres turn radiation into weather, kinetic energy.  radiation can also induce phase changes, e.g., comets.  maybe some of these processes have equilibrated.  if none or not much of these processes are happening, is a planet dust?

[bccrjwsd] highest drama in sports

baskeball: the team down by 3 is facing elimination (e.g. single elimination tournament, or best of N series), and time is running out.  they make a 3-point shot as time runs out, tying the game.  this could happen at the end of regular time or at the end of an overtime period.  the team goes on to win not only the game but the season.  most dramatic: it is the final championship game (game 7 if best of 7).  it's a little unfortunate that the moment of highest drama only results in a tie, but this is unavoidable.  if the most dramatic action were to instantly win, then there's the nagging possibility that an easier route, tie first then win in overtime, was possible.

baseball: not a single moment, not a single player, but a team effort.  there is no clock in baseball.  largest deficit closed during the last out of the bottom of a possibly final inning (typically bottom of 9th, but could be bottom half of an extra inning).  outcome of the inning could be a tie (in which case (another) extra inning), or a walk-off victory.  for the latter, the measure of drama is only the deficit needed to tie, not the potentially up to 3 extra runs (grand slam) to win.

unlike basketball, for baseball it does not matter what game this record-deficit overcoming occurred, or what the ultimate outcome of the game or season was.  a larger deficit is always more dramatic than a smaller one, even if the former happened in a game that "doesn't matter" and the latter in an important game.  of course, when comparing the same deficit, more important games are more dramatic.

does it matter the number of players on base at the start of the 2-out rally?